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Stevan Gajović

Publications and source records attributed to Stevan Gajović.

12 recordsLinked to original sources
arXiv2026

The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny

We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic numbers. We determine these densities by calculating the corresponding $p$-adic volume integrals and analyzing certain modular curves. Additionally, we explore the case of $\ell$-torsion for $\ell>3$ prime.

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arXiv2026

On $p$-adic denseness of quotients of values of integral forms

Given $A\subseteq \Z$, the ratio set or the quotient set of $A$ is defined by $R(A):=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of values attained by an integral form. For a given form, we investigate whether this happens for all but finitely many $p$. We consider the more general question when forms have coefficients in Dedekind domains, and, under certain conditions, we prove that the analogous statement holds. However, we also give examples of integral forms for which the answer is negative, and it is crucial that the degree of such forms is composite. We conjecture that we cannot find such examples with forms of prime degree having sufficiently many variables, which is indeed the case when the degree is 2, 3, or 5. Our innovation is to consider the problem in terms of varieties defined by forms, thereby using the tools from algebraic geometry, making their first appearance in this setting. We construct integral forms such that the ratio set of its values is dense in at least one $\Q_p$ but only in finitely many of them.

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arXiv2026

Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our method, over $\mathbb{Q}$, we prove that there exist infinitely many non-isomorphic hyperelliptic curves of genus two, whose Jacobians have rank equal to a fixed number between $1$ and $11$, genus three and four curves with rank between $1$ and $4$, and genus five and six with rank between $1$ and $3$.

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arXiv2026

Pythagoras numbers for infinite algebraic fields

We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.

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arXiv2025

Linear quadratic Chabauty

We present a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the $p$-adic height to a linear function; we can then express this restriction in terms of holomorphic Coleman integrals under the standard quadratic Chabauty assumption. Then we use this linear relation to extract the integral points on the curve. We also generalize our method to integral points over number fields. Our method is significantly simpler and faster than all other existing versions of the quadratic Chabauty method. We give examples over $\Q$ and $\Q(\sqrt{7})$.

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arXiv2025

Arithmetic-geometric mean sequences over finite fields $\mathbb{F}_q$, where $q\equiv5\pmod{8}$

Arithmetic-geometric mean sequences were already studied over real and complex numbers, and recently, Michael J. Griffin, Ken Ono, Neelam Saikia and Wei-Lun Tsai considered them over finite fields $\mathbb{F}_q$ such that $q \equiv 3 \pmod 4$. In this paper, we extend the definition of arithmetic-geometric mean sequences over $\mathbb{F}_q$ such that $q \equiv 5 \pmod 8$. We explain the connection of these sequences with graphs and show the properties of the corresponding graphs in the case $q \equiv 5 \pmod 8$.

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arXiv2024

Computing p-adic heights on hyperelliptic curves

We describe an algorithm to compute the local Coleman-Gross p-adic height at p on a hyperelliptic curve. Previously, this was only possible using an algorithm due to Balakrishnan and Besser, which was limited to odd degree. While we follow their general strategy, our algorithm is significantly faster and simpler and works for both odd and even degree. We discuss a precision analysis and an implementation in SageMath. Our work has several applications, also discussed in this article. These include various versions of the quadratic Chabauty method, and numerical evidence for a p-adic version of the conjecture of Birch and Swinnerton-Dyer in cases where this was not previously possible.

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arXiv2023

Reverse engineered Diophantine equations

We answer a question of Samir Siksek, asked at the open problems session of the conference ``Rational Points 2022'', which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set $S$ of perfect integer powers, using Mihăilescu's theorem, we construct a polynomial $f_S\in \Z[x]$ such that the set $f_S(\Z)$ contains a perfect integer power if and only if it belongs to $S$. We first discuss the easier case where we restrict to all powers with the same exponent. In this case, the constructed polynomials are inspired by Runge's method and Fermat's Last Theorem. Therefore we can construct a polynomial-exponential Diophantine equation whose solutions are described in advance.

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arXiv2021

Cubic and quartic points on modular curves using generalised symmetric Chabauty

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell--Weil group of its Jacobian.

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arXiv2020

Curves with sharp Chabauty-Coleman bound

We construct curves of each genus $g\geq 2$ for which Coleman's effective Chabauty bound is sharp and Coleman's theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman's bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman's theorem.

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